{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2014:ZKLOIZMX2IMU3CLCHN7VDUBX5E","short_pith_number":"pith:ZKLOIZMX","canonical_record":{"source":{"id":"1412.6022","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-18T19:30:06Z","cross_cats_sorted":[],"title_canon_sha256":"4b604479ff4263952559499ab9320b759619e9a4e9132933f4b461e01bce56ac","abstract_canon_sha256":"8a12f40bfb97df5464a7c3afaee9d1a22f7ddc4d076ea680a6b015a6c7db1948"},"schema_version":"1.0"},"canonical_sha256":"ca96e46597d2194d89623b7f51d037e917e9520d39fd943bc78f500b83824b05","source":{"kind":"arxiv","id":"1412.6022","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.6022","created_at":"2026-05-18T01:21:19Z"},{"alias_kind":"arxiv_version","alias_value":"1412.6022v2","created_at":"2026-05-18T01:21:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.6022","created_at":"2026-05-18T01:21:19Z"},{"alias_kind":"pith_short_12","alias_value":"ZKLOIZMX2IMU","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZKLOIZMX2IMU3CLC","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZKLOIZMX","created_at":"2026-05-18T12:28:59Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2014:ZKLOIZMX2IMU3CLCHN7VDUBX5E","target":"record","payload":{"canonical_record":{"source":{"id":"1412.6022","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-18T19:30:06Z","cross_cats_sorted":[],"title_canon_sha256":"4b604479ff4263952559499ab9320b759619e9a4e9132933f4b461e01bce56ac","abstract_canon_sha256":"8a12f40bfb97df5464a7c3afaee9d1a22f7ddc4d076ea680a6b015a6c7db1948"},"schema_version":"1.0"},"canonical_sha256":"ca96e46597d2194d89623b7f51d037e917e9520d39fd943bc78f500b83824b05","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:21:19.774234Z","signature_b64":"QJnJgozkPCE0Up+FleDl7Q1/TrYdyFpGEGgwo215a4uAaQow8SOH8F+JHZeyps/UyvWXxxoIOHcGt0XZFg6CBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ca96e46597d2194d89623b7f51d037e917e9520d39fd943bc78f500b83824b05","last_reissued_at":"2026-05-18T01:21:19.773762Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:21:19.773762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1412.6022","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:21:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"u2gW/Lhc5+Rs6IJnuJSnk8w9TZBXkjrybcWkm742/JfM7hB07YtOc05Sp+Y/oEMomEmgMMwizh2Fwuh0CVVFCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T03:27:50.790587Z"},"content_sha256":"940ed60e1cfcfcf12f68a62541d3b8a7a65353dc8084598c384720e324d77d73","schema_version":"1.0","event_id":"sha256:940ed60e1cfcfcf12f68a62541d3b8a7a65353dc8084598c384720e324d77d73"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2014:ZKLOIZMX2IMU3CLCHN7VDUBX5E","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Ground states of nonlinear Schr\\\"odinger equations with sum of periodic and inverse-square potentials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Jaros{\\l}aw Mederski, Qianqiao Guo","submitted_at":"2014-12-18T19:30:06Z","abstract_excerpt":"We study the existence of solutions of the following nonlinear Schr\\\"odinger equation \\begin{equation*}\n  -\\Delta u + \\Big(V(x)-\\frac{\\mu}{|x|^2}\\Big) u = f(x,u)\n  \\hbox{ for } x\\in\\mathbb{R}^N\\setminus\\{0\\}, \\end{equation*} where $V:\\mathbb{R}^N\\to\\mathbb{R}$ and $f:\\mathrm{R}^N\\times\\mathbb{R}\\to\\mathbb{R}$ are periodic in $x\\in\\mathbb{R}$. We assume that $0$ does not lie in the spectrum of $-\\Delta+V$ and $\\mu<\\frac{(N-2)^2}{4}$, $N\\geq 3$. The superlinear and subcritical term $f$ satisfies a weak monotonicity condition. For sufficiently small $\\mu\\geq 0$ we find a ground state solution as "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.6022","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:21:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uFEYsWFiNaHZRyJKDRUlL7srdLTvs9ZZ//9gBghR9Mtt2Fymz4ORM3hyFbpLTTQc6V27ZbIT65ngQXEl195nAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-03T03:27:50.790933Z"},"content_sha256":"b36083d4817d07367b28362235057dbbb53c067a43ed760991c53cfe06893aff","schema_version":"1.0","event_id":"sha256:b36083d4817d07367b28362235057dbbb53c067a43ed760991c53cfe06893aff"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZKLOIZMX2IMU3CLCHN7VDUBX5E/bundle.json","state_url":"https://pith.science/pith/ZKLOIZMX2IMU3CLCHN7VDUBX5E/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZKLOIZMX2IMU3CLCHN7VDUBX5E/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-03T03:27:50Z","links":{"resolver":"https://pith.science/pith/ZKLOIZMX2IMU3CLCHN7VDUBX5E","bundle":"https://pith.science/pith/ZKLOIZMX2IMU3CLCHN7VDUBX5E/bundle.json","state":"https://pith.science/pith/ZKLOIZMX2IMU3CLCHN7VDUBX5E/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZKLOIZMX2IMU3CLCHN7VDUBX5E/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:ZKLOIZMX2IMU3CLCHN7VDUBX5E","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"8a12f40bfb97df5464a7c3afaee9d1a22f7ddc4d076ea680a6b015a6c7db1948","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-18T19:30:06Z","title_canon_sha256":"4b604479ff4263952559499ab9320b759619e9a4e9132933f4b461e01bce56ac"},"schema_version":"1.0","source":{"id":"1412.6022","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1412.6022","created_at":"2026-05-18T01:21:19Z"},{"alias_kind":"arxiv_version","alias_value":"1412.6022v2","created_at":"2026-05-18T01:21:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1412.6022","created_at":"2026-05-18T01:21:19Z"},{"alias_kind":"pith_short_12","alias_value":"ZKLOIZMX2IMU","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_16","alias_value":"ZKLOIZMX2IMU3CLC","created_at":"2026-05-18T12:28:59Z"},{"alias_kind":"pith_short_8","alias_value":"ZKLOIZMX","created_at":"2026-05-18T12:28:59Z"}],"graph_snapshots":[{"event_id":"sha256:b36083d4817d07367b28362235057dbbb53c067a43ed760991c53cfe06893aff","target":"graph","created_at":"2026-05-18T01:21:19Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We study the existence of solutions of the following nonlinear Schr\\\"odinger equation \\begin{equation*}\n  -\\Delta u + \\Big(V(x)-\\frac{\\mu}{|x|^2}\\Big) u = f(x,u)\n  \\hbox{ for } x\\in\\mathbb{R}^N\\setminus\\{0\\}, \\end{equation*} where $V:\\mathbb{R}^N\\to\\mathbb{R}$ and $f:\\mathrm{R}^N\\times\\mathbb{R}\\to\\mathbb{R}$ are periodic in $x\\in\\mathbb{R}$. We assume that $0$ does not lie in the spectrum of $-\\Delta+V$ and $\\mu<\\frac{(N-2)^2}{4}$, $N\\geq 3$. The superlinear and subcritical term $f$ satisfies a weak monotonicity condition. For sufficiently small $\\mu\\geq 0$ we find a ground state solution as ","authors_text":"Jaros{\\l}aw Mederski, Qianqiao Guo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-18T19:30:06Z","title":"Ground states of nonlinear Schr\\\"odinger equations with sum of periodic and inverse-square potentials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.6022","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:940ed60e1cfcfcf12f68a62541d3b8a7a65353dc8084598c384720e324d77d73","target":"record","created_at":"2026-05-18T01:21:19Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"8a12f40bfb97df5464a7c3afaee9d1a22f7ddc4d076ea680a6b015a6c7db1948","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AP","submitted_at":"2014-12-18T19:30:06Z","title_canon_sha256":"4b604479ff4263952559499ab9320b759619e9a4e9132933f4b461e01bce56ac"},"schema_version":"1.0","source":{"id":"1412.6022","kind":"arxiv","version":2}},"canonical_sha256":"ca96e46597d2194d89623b7f51d037e917e9520d39fd943bc78f500b83824b05","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ca96e46597d2194d89623b7f51d037e917e9520d39fd943bc78f500b83824b05","first_computed_at":"2026-05-18T01:21:19.773762Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:21:19.773762Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QJnJgozkPCE0Up+FleDl7Q1/TrYdyFpGEGgwo215a4uAaQow8SOH8F+JHZeyps/UyvWXxxoIOHcGt0XZFg6CBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:21:19.774234Z","signed_message":"canonical_sha256_bytes"},"source_id":"1412.6022","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:940ed60e1cfcfcf12f68a62541d3b8a7a65353dc8084598c384720e324d77d73","sha256:b36083d4817d07367b28362235057dbbb53c067a43ed760991c53cfe06893aff"],"state_sha256":"5c1f2c06e0e16badc14f7ea82453f9c0df6fa9d7defad897d66eea6db8ce5416"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RP7dhTGczv9rxKei9+mp5Lc5pOcr6FIB5E6LmhU+RZ2lNfd4oMkY4aEHgkCwt5+Tos7/RgPjhwxjgaTcLpdSBQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-03T03:27:50.792948Z","bundle_sha256":"7c299208ac99a4102d5971a0368a4cfb63f910cc4105244cc4608813ac879f1f"}}